Combine all the ingredients in a small bowl for the garlic mayo and refrigerate until ready to use.
Divide the meat into four equal portions. Form each portion loosely into a disk shape, making a deep impression in the centre with your thumb. Season both sides of each burger to taste with salt and pepper.
Prepare and preheat a grill. Cook the burgers until slightly charred and cooked through, about 5 minutes per side.
Remove the burgers from the heat and add the brie cheese. Tent with aluminum foil and allow the burgers to rest for 5 minutes, or until the cheese is melted.
Top each burger with arugula and caramelized onions. Spread garlic mayo on each burger bun and sandwich the burgers.